Ch. 13

Question 1

What to do when the assumptions are not true?

Answer 1

1. Ignore the violations
2. Transform the data
3. Use nonparametric method
4. Use permutation test

Question 2

When is data not likely to be normal, graphically?

Answer 2

(pg. 371)
When distribution is strongly skewed or strongly bimodal, or has outliers

Question 3

Normal Quantile Plot

• What is it?

Answer 3

Compares each observation in the sample w/ its quantile expected from the standard normal distribution

If it is normally distributed, points should roughly be in a straight line.

Question 4

Shapiro-Wilk test

What is it?

Answer 4

Shapiro-Wilk test evaluates goodness of fit of a normal distribution to a set of data randomly sampled from population (null hypothesis is that it is normal)

Question 5

Recommended methods of evaluating assumption of normality?

Answer 5

• Can do statistical test (ex. Shapiro-Wilk test), but has false sense of security
• IDEALLY should use graphical methods & common sense to evaluate (frequency distribution histograms, normal quantile plots)

Question 6

Robust

Def’n?

Answer 6

A statistical procedure is robust if the answer it gives is not sensative to violations of the assumptions of the method.

Question 7

Normal Approximation - Ignoring violations

• For what types of tests?
• Threshold of “ignorance”?

Answer 7

• For tests that use the mean (robustness due to the central limit theorem)
• Ignorance threshold rule of thumb is n > 50(ish)
  
  • Also need to consider the shape of the distributions; skew needs to be similar, and no outliers. See pg. 376

Question 8

When can assumptions of equal standard deviations can be ignored?

Answer 8

• If n > 30 for each group, and n is similar for both groups (approximately), then SD can be ignored even w/ greater than 3x difference
• When:
  • n is not approximately equal

Question 9

Data transformation def’n

Answer 9

Data transformation changes each measurement by the same mathematical formula

Question 10

Purpose of a transform?

Answer 10

• To attempt to make SD more similar and to improve the fit of the normal distribution to the data.

NOTE: This tranform will affect all the data AND the hypotheses equally; i.e. everything gets tranformed the same way. Also, can’t just do ln[s] for example; have to re-calculate starting with the mean of all the logs.

Question 11

Examples of possible transformations?

(indicate top 3)

Answer 11

• Log transform
• Arcsine transform
• Square-root transform
• Square transform
• Antilog transform
• Reciprocal transform

Question 12

Log tranform - what does it do?

Answer 12

Converts each data point to its logarithm

Ex. Y = ln[X]

Question 13

What to do if you want to try log transform but data has zero?

Answer 13

Try Y’ = ln[Y + 1]

Question 14

Log transform - When is it useful?

Answer 14

• When measurements are ratios or products of variables
• When frequency distribution is right skewed
• When group that has the larger mean (when comparing 2 groups) also has the larger SD
• When data spans several orders of magnitude

See pg. 378 for details

Question 15

Arcsine tranformation

• What does it look like?
• What is it needed for?
• How does it fix things?

Answer 15

• p’ = arcsin[sqrt(p)]
• Used for proportions
• Makes it closer to a normal distribution and also makes SD’s similar

Note: convert percentages into decimal proportions first b4 application

Question 16

Square-root tranformation

• What does it look like?
• What is it needed for?
• How does it fix things?

Answer 16

• Y’ = sqrt( Y + 1/2)
  
  • sometimes Y + 0 or Y + 1)
• Used for count data
• Effect similar to log; makes SD similar for comparisons where the larger mean is also the the higher SD
  
  • If effects same as log; use either or

Question 17

Square tranformation

• Transform?
• When to use?

Answer 17

• Y’ = Y^2
• When frequency distribution is skewed left
  
  • Only usable if all Y have same sign
  • If all negative, try multiplying all by -1 first

Question 18

Antilog transform

• Transform?
• When to use?

Answer 18

• Y’ = e ^Y
• Use when square transform doesn’t work on left-skewed data

Question 19

Reciprocal tranform

• Transform?
• When to use?

Answer 19

• Y’ = 1/Y
• When data is skewed right
  
  • Only usable if all Y have same sign
  • If all negative, try multiplying all by -1 first

Question 20

Calculating CI with transform?

Answer 20

Use the transformed values, then once range is calculated, it is best to convert BACK into original scale by back-transform (i.e. invert the transformation).

See pg. 382

Question 21

Valid transforms . . .

Answer 21

• Require same transform applied to each individual
• Have 1-to-1 correspondence to OG values
• Have monotonic relationship w/ OG values (large values stay larger)

Question 22

Nonparametric methods - def’n

Answer 22

Nonparametric methods make fewer assumptions than standard parametric methods do about the distribution of variables

• Achieves this by ranking the data
• AKA “distribution-free” methods”

Question 23

Ranking Data points

Answer 23

• Rank in smallest to largest.
• Ties are resolved by averaging what the ranks would be if it was sequential, then assigning the next rank up to the next largest individual
  
  • ex. 5 (Rank: 1), 6, 6, 8
  • 6’s would be 2 and 3, so average 2 + 3, = 2.5 (“midrank”)
  • So the rank of 6 is 2.5 and 2.5, and 8 is the next rank after the highest one (3), i.e. 8 is ranked 4

See pg. 391

Question 24

Sign-test

• What is it?
• What is its parametric equivalent(s)?

Answer 24

• Compares the median of a sample to a constant specified in the null hypothesis. Makes NO assumptions about distribution of measurement in population
• Equivalents are one-sample or paired t-tests

Question 25

Problems with sign-test?

Answer 25

* Low power compared to t-test * likely to NOT be able to reject a null * Impossible to reject null if n is less than/equal to 5 * Requires large sample sizes * Requires omission of data that exactly equal the hypothesized null median

Question 26

Mann-Whitney U-test * What does it do? * Parametric equivalent?

Answer 26

* Compares distributions of 2 groups. Does not require as many assumptions of 2 sample t-test * if distributions same, will test centeral tendencies using ranks (medians, means) * Replaces two-sample t-test

Question 27

Assumptions of the Mann-Whitney U-test?

Answer 27

* Both samples are random samples * Both populations have the same shape of distribution

Question 28

Effect of assumptions on Type 1 Error rates?

Answer 28

* If assumptions of a given test (parameteric or nonparametric) are met, then prob[Type 1 Error] = a * when not met, type 1 error becomes larger than a * this is why parametric tests are not used when assumptions are violated

Question 29

Effect of assumptions on Type 2 Error rates?

Answer 29

* Parametric tests use ranks, thus uses less info * less info means **less power**, less power means **lower probability of rejecting false** null hypotheses (i.e. increase Type 2 error) * Reduced power of nonparametric tests is irrelevant when parametric test assumptions are violated

Question 30

Relative powers of Mann-Whitney U-test and sign test

Answer 30

At best (i.e. with large samples), * Mann-Whitney is about 95% as powerful as two-sample t-test * sign test has about 64% power of t-test (much lower power) * thus is a last resort method Power decreases with smaller sample sizes.

Question 31

Permutation test - def'n

Answer 31

Generates a null distribution for the association between two variables ("measures of association") by repeatedly and randomly rearranging the values of one of the two variable in the data. \*Randomization is without replacement Sometimes called "randomization test"

Question 32

When can permutation tests be used?

Answer 32

Can be done for ANY test of association between two variables (categorical and numerical, numerical x2, categorical x2)

Question 33

Assumptions of the Permutation Test? Power of the Permutation Test?

Answer 33

* Must be a random sample * For permutation tests comparing means and medians, the distribution **must have the same shape in every population** * Robust to violations in this when sample sizes are large, more than Mann-Whitney * with small sample size, has less power than parametric tests (but still more powerful than Mann-Whitney) * Similar power to parametric when n is large